It is an interesting perspective moving out of the classroom into a coaching position. I have had more face-to-face teacher math conversations this year than ever in my career and it is wonderful. This position also lets me take a step back from the daily lesson planning and think about things I see across all of the grade levels. Most times, my thoughts are about the trajectory of mathematical ideas, however over the past couple of weeks I find myself thinking about two things I saw as norms when I taught, but now wonder more about…
1 – Is there a such thing as an addition, subtraction, multiplication or division problem?
I am sure we all can relate to the stories of students struggling with story problems. We see them be successful with Notice/Wonders and 3-Act Math tasks, however when given a story problem some “number grab” and compute without thinking about reasonableness. Why? While I think there are many factors at play here, I have another theory that has led me to question problem types. I could be completely off, but as I look through the curriculum and think about the progression in which I taught in 5th grade, I wonder if there is something to teaching “types of problems” within a unit. For example, in Unit 1, Investigation 1 could be my multiplication lessons while Investigation 2 could be my division lessons. While we don’t explicitly say, “this is how you solve a multiplication problem” and we explore various strategies to make connections between the operations, the header of the activity book pages say things such as, “Division Stories” or “Multiplication Stories.” Also, the majority of the work that week is the specific operation and applications.
From there I began to wonder, is there really a such thing as a specific operation problem? I would think that any division story could also be thought of as a multiplication problem. Do we lead students to think there are certain types of problems even if we make clear all of the strategies to find solutions? I love how CGI talks about problem types and wondering why more curriculum are not set up that way instead of keying students into operation-specific problems?
I asked some 4th graders about this exact idea. I gave them some multiplicative compare problems and asked them if they thought about each as multiplication, division or both. Then we talked about why.
2 – What makes students attach meaning to a vocabulary word? Do they need to?
Every year in 5th grade, I was confident that all of my students could find area and perimeter of rectangles. However, I was also confident that there would also be a handful of students who could find area and perimeter but didn’t know which was which. After much work with area and perimeter, they would have it by the end of the unit, but did they remember when they got to 6th grade? I am not sure.
Now, seeing all of the work they are doing with this beginning in 3rd grade, and talking to 3rd and 4th grade teachers who are seeing the same thing, I am left wondering why this is? What makes students attach meaning to vocabulary? This question is then followed by the my very next question…when do they have to?
I wonder if students should ever be given a problem where the context would not allow the students to figure out which one, area or perimeter, the problem was asking. For example, if Farmer Brown is buying fencing he would need the perimeter where if he was buying something to cover a piece of ground, it would be area. Should we ever give them just naked perimeter or area problems with no context where knowing the meaning of the word impacted their ability to solve it?
And then, after they do all of this work with both measurements, why do they forget which word is which year after year? I know the teachers do investigations with the work and use the vocabulary daily during the unit, as I did, but students still don’t hold on to it. What makes it become part of their vocabulary? Is it just too long between when they use it? Is it
These are just two things I am wondering about….
The word “area” is fairly common in everyday language, but WHO talks about perimeters. “How far round is it” is the usual description. There are too many jargon words in elementary math. How about “properties of operations”, divisor,dividend and quotient, multiplicand, scalene, ..more..
I just wondered whether the heading on the illustrated worksheet you show is for the kids benefit or for teachers filing needs!!!!!!!
I know Howard, I often have so many curiosities about the importance of certain vocabulary words and wonder if oftentimes testing is why teachers feel such a need to focus on them?
I agree, it looks partly like testing that skews us towards vocabulary. It’s easier to test.
I also imagine – the equivalent is certainly true over here – that when some teachers read “Solve real world and mathematical problems involving perimeters of polygons”, they think that children need to be learning the words perimeter and polygon. But – correct me if I’m wrong – the standard is asking primarily for something to be understood and done, and not so much that the words to be learnt.
Great wonderings! I particularly connected to your first question. I think we do a lot of hand holding and leading kids to a particular strategy or operation when we assess or even assign problems. I know I am guilty of this as it is how I naturally think(really the way I was taught myself) about something in math. In other words, I try to categorize a problem or label it so that students can understand and have a context for solving it. So many of my students simply cling to the algorithm without truly understanding the math, which is really the beauty of the whole thing. I am going to think about rephrasing some of the problems I give students so that the operation to be used is not so obvious. It would be great if you wrote a blog post addressing this issue.
Thanks so much for your comment Elisa, it really is such an important thing to think about in our teaching. I know I mix it up with number talks and such, but some of our curriculum units are labeled “multiplication” and wonder if the students, after a week or two of that unit just assume every problem will be multiplication and that leads to our “number grabbers”?
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Good point! That’s why I like some of the problems I’ve been using on the nrich website. It gets kids out of that “number grabber” frame of mind.
In your role as a math coach, visiting multiple grade levels, you’ll find yourself wondering lots of things you never wondered as a classroom teacher! Your observations about perimeter and area are great examples. There are topics that come up, year after year, and so many kids continue to struggle that I wonder, “What are we doing wrong?” i wish I had an answer.
A former supervisor of mine who now teaches at Hunter College talks about the possible confusions that can occur for kids when we teach things like perimeter and area together. Perhaps we’d be better off teaching about perimeter, waiting as several units go by, then tackling area.
Vocabulary is tricky too. We’re always in such a rush to give formal names to things, as if the fact that kids can parrot the vocabulary back to us is a big deal. Isn’t it better, at least in the beginning when the concepts are being formed, for kids to invent their own vocabulary and names for things? Names that make sense to them?
I think we also need to think of the developmental appropriateness of concepts. We wind up doing damage by rushing and introducing things too early, then wind up having to spend time trying to undo all the misunderstanding.
Anyway, great post! Lots to think about.
I agree with you and Joe and Howard about vocabulary. And that phenomenon of learning a technical term only to forget it a year later is so familiar, it ought to have a name. Can we agree to call it “frugalcy”? Likewise with the problem of learning two of them at the same time, only to be not quite sure which is which after a year: that needs a name. I propose “eleseris”. Try and use these over the next week when discussing this. You won’t mind if I check up in a year whether you remember them and know which is which?
(And of course, there is a relationship between frugalcy and eleseris. The bigger the frugalcy of the separate words, the bigger the eleseris with the two combined.)
I completely agree with you all and Simon that is a perfect vocabulary exercise! So the big question, that is really interesting to me, is what really makes a word stick with us?
But seriously… I introduce vocabulary a bit slower than lots of teachers (we’ve just been doing fractions without using the words denominator and numerator; I tend to not focus on types of triangles or quadrilaterals when we’re looking at shape; I don’t usually use terms like subtrahend and minuend). But even so, I am still probably guilty of vocabulary overload. I think the ideas of factors and prime numbers, and indeed prime factors are key, so I give it time; but I bet there are some students who are not quite sure of those words this term. Maybe, like Joe, I should hold back on “prime” – I don’t know…
Recently as well, Miguel asked what the square root symbol means. I think the students can understand this quite well, and it seems odd to just talk about the symbol without giving it a name. But of course, this is normally left to a lot later. And yet I like what Joe says about rushing and introducing things too early. Am I being greedy, wanting to show too much, too soon? I want everyone to be included. Again, I don’t know…
I know! I like to offer the vocabulary when the students feel like they need it to make their explanations easier. So, instead of saying, “The distance around the shape,” eventually I think it makes it easier to say the perimeter. But definitely all after they have had a lot of experience with the work and never in a “memorize this”-kinda way.
I suppose we need to be sensitive to each child’s needs. Miguel is curious about something. Even though it’s something that is normally left to later, he wants to know about it now. So I think he deserves an explanation.
Exploring the nexus of math and vocabulary, and symbols and their meanings, is interesting, especially viewed through the lens of the question: Why do we have to repeatedly teach the same thing over and over and over again and still many kids are confused? It seems so frustrating. Is it us? Is it them? Is it purely the concept? The names we give to the concept? I wish I could go back in time and pinpoint the moment I really, truly understood the difference between area and perimeter!
Is it us? Is it them? Is it purely the concept? The names we give to the concept? <—– YES, this is the exact thing I am thinking about right now! And then I move to Why do they really even need that work without a context that would allow them to know which one is which?
There are so many experimental psychology studies of memory; there must be some with a bearing on this – I just don’t know them. They would of course have to reflect how our classrooms work, which is I suppose where these things usually lose validity.
So, a week working on let’s say area and perimeter, one group with and another without the words area and perimeter, how well after a year would the students be able to find areas and perimeters of rectilinear figures (obviously without the words being used in the case where they didn’t learn them; possibly two conditions, with and without vocabulary, for where they did learn the words)? How would putting a time between learning about area and perimeter affect this? (This is what I do by the way.)
Maybe someone who reads this knows of studies like this?
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How about making some flexible polygons and drawing round them on squared paper.
Shouldnt be too long before the students see that “total length of sides” and “estimated number of squares enclosed” do not have much in common.
Howard, you raise an interesting issue with squared paper. I’ve often felt that the paper might add to the confusion because the kids have a hard time separating the idea of perimeter as a measure of length when they also count boxes. I’m not sure they understand that when finding perimeter they are really looking at the sides of the boxes, not the boxes themselves. Here’s a thought: maybe we should say “fence” instead of perimeter. Like: “Find the fence of this rectangle.” Most every perimeter problem has something to do with a fence anyway!
Regarding confusing symbols, here was my attempt to make some sense out of the long division “house”.
I like that. There is also the repeated subtraction approach, which is almost the same. I am following your blog now.
I actually spent 2015 doing some thinking on this exact topic (#2)and would love to get some more ideas on how to best support my students’ understanding.
I had just started at a new school with a new year level group 2015 and after coming to the end of a perimeter/area unit I realised that some students were still not always able to remember how to answer correctly. I knew I hadn’t taught absolutely as I would have wished but I had worked with them on lots of hand on experiences as well as unpacked the key vocabulary so I was a little surprised. It was mainly a problem with the least able in my class but I felt frustrated that I hadn’t met their needs.
On reflection I decided I should have gone slower during the process of discovering and using formulae and definitely needed to have monitored understanding more throughout the unit but it didn’t feel like that was the only answer. I knew I would need to make opportunities to review perimeter and area throughout the year with this group.
I did a little looking around, as you do, and came across some work around mastery learning that discussed similar ideas shouldn’t be taught at the same time – like multiples and factors are similarly often confused.
I have also been doing some thinking around a less linear year overview so started to think about how I could also inbed perimeter and area more within the numeracy strategies. As I was teaching a lower grade the same topic later in the year I decided to make an effort to instead of ‘teaching perimeter’ rather use it as a context within a unit on addition and subtraction strategies. In particular we looked at compatible numbers and place value strategies. Then later in the year within the multiplicative thinking unit I brought in area etc. I did also include problems at that time like maximising area for a given perimeter so as to (I hoped) further review and solidify their perimeter understanding.
I had also done some reading on spacing/interleaving so made efforts to come back to these ideas throughout the year where possible. It also struck me when we were focussing on measurement the struggle students have using the correct measuring units. So throughout the year I tried to slip measuring tasks in wherever I could, doing statistics great, we also did investigations into our relative body parts a la the vitruvian man.
So I have a lot swirling around in my head that I want to work on such as – integrating mathematical ideas more within problem solving, more peer sharing and further enhancing the level of math discourse, better maintaining already taught ideas throughout the year, giving students enough time to construct and make deeper connections, using formative assessment more within my planning cycle, having students be more aware of their learning needs…
I find it really difficult though with the students who need so many more experiences and work to develop their number sense and how they often need more time to really understand a concept and how can I best meet their needs whilst still challenging my most able.
I’m doing what I hope will work but it kind of feels like we’re in the classroom but do we really know if what we’re doing is on point? And if someone knows more I would love to be told.
I’ve been working with a group of teachers using CGI in my district. I love this book! It’s really helping the teachers think differently about how they introduce story problems and how they think about the mathematical thinking their students are engaging in on a daily basis.
Stupid question section: Does a circle have a perimeter?